Rocketplanes update

Anywhere in an hour for $100K. Can hypersonics compete?

Code for the calculations in this post can be found here.

Bean pointed out an error in my original post on rocketplanes. You can’t just throttle a rocket to a certain delta-V and expect it to remain along an orbital path. This completely changes the conclusions of the original post.

Let’s use the equations for a projectile flying over a planetary surface. Aimed at 45 degrees, with instantaneous acceleration and no atmosphere, the distance traveled is:

\(d = \frac{V^2}{g -\frac{V^2}{2R_e}}\)

Where R_e is the radius of the Earth. It’s more useful for our purposes to write V in terms of d:

\(V = \sqrt{\frac{d \cdot g}{1 + \frac{d}{2Rₑ}}}\)

How much time does the trip take? Wikipedia has this awful expression:

The V-tilde is the ratio between V and the first cosmic, sqrt(g*R_e).

\(\tilde{v} = V / \sqrt{g R_e}\)

Let’s focus on that arcsin term. When I plug in 45 degrees on Wolfram Alpha, this is what I get (where x is v-tilde):

V-tilde can only range from 0 to 1 before the assumptions of the equation break down. The orange line is my cheeky linear approximation. Since it upper-bounds the time, this is a simple and conservative approximation; appropriate for an optimistic cost-benefit analysis.

Using 1.5*V-tilde we get:

\(t = \frac{V\sqrt{2}}{g} \cdot \frac{1}{2-\tilde{V}^2} \cdot (1 + \frac{\sqrt{2}}{\tilde{V}\sqrt{2-\tilde{V}^2}}) \cdot \frac{3\tilde{V}}{2}\)

Fuel is a major cost of the flight, to boost to a certain delta-V requires:

Where m_x is the mass of fuel required, m_p is the mass of the rocket plus the mass of the people, I_sp is the specific impulse of the rocket and g is Earth’s gravitational acceleration. And you need to slow back down when you reach your destination, doubling the fuel required.

So for a given trip distance, we know how fast we have to fly, how long the flight will take, and how much fuel that requires. Below is a table for different distances, assumptions in the appendix.

So rocketplanes can get you anywhere on Earth in about an hour at a fuel cost of around $20K. I expect things like repairs, capital costs, staff, and insurance to contribute significantly beyond fuel costs. So the real number would be perhaps 5-10x larger. $100K seems like a good ballpark.

The flight times are so short that other things determine the duration of the trip. You’ve got to change into a jumpsuit, take a mandatory bathroom break, get weighed, and have your vitals checked. It takes a time to accelerate and decelerate. The launch and landing pads are remote, so there’s added commute time on each end. These can add hours to the trip.

Hypersonics are competitive

Rockets would have to compete with super- or hypersonic aircraft companies like Astro mechanica, Hermeus, Venus Aerospace, Boom, Spike, and others. These designs don’t need to carry their oxygen on board, reducing fuel mass by 5x since they don’t need LOX. These planes can also take off from the same runways we have today and can achieve speeds of Mach 1-5. How do they compare?

Claude gave me this helpful chart of the distances between cities that can act as hubs:

The typical trip is about 10,000 km. A Mach 5 plane can cover that in … 97 minutes. When you factor in the closer location of airports versus launch pads, the hypersonic plane is probably faster overall. Add in the fact that hypersonics use much less fuel and they start to look competitive with rockets. On the other hand, hypersonic planes don’t exist yet and would only carry a few passengers per flight. The winner will come down to capital cost, number of passengers, and launch cadence more than fuel.

On balance, I think hypersonics will outcompete rockets for the high-speed market. People’s time just isn’t valuable enough to shave a few minutes off a flight. I have to sleep anyways, and I can do that on the plane rather than my hotel¹. But there might be a niche for risk-loving CEO’s that need to get from Rio to Tokyo fast and at any price²³. For shorter trips, there are plenty of ways to make traditional airplanes cheaper, faster, and more comfortable.

The future looks like a patchwork of different transportation modes. People take airplanes within their continent and hypersonics for far-flung vacations. But anytime there’s a geopolitical crisis, rockets full of diplomats might start flying.

Appendix

• For the rocket equation, I based numbers off of the SpaceX Raptor engine, with an I_sp of 350 seconds in atmosphere.

• At Bean’s suggestion, I upped the mass per passenger to 500 kg total. That’s 75 kg of body mass per person and 425 kg of rocket hardware per person.

• The cost of fuel C_fuel has conflicting estimates, this article claims that methane costs $9/kg, but looking at natural gas spot prices, I get something more like $0.1-$0.2/kg. I’m going to stick with the higher number for now. Note that liquid oxygen is much cheaper, and methane is only 20% of the overall propellant mass.

1. Perhaps with hypersonics you could throttle the speed based on the distance so that every trip is a redeye that takes 6 hours.

2. Sounds like a good premise for a movie actually.

3. Rocketplanes might also find application for moving critical supplies like troops, military equipment, and disaster aid.